Why having a good uv map is important?
In my answer I will only cover the importance of good UV layouts for real-time graphics engines, because it's the only topic I have enough experience in to write about. 

Importance in Real-time Graphics Engines
Besides giving artists the advantage to hand-paint models, good UV maps are very important for any type of real-time graphics engine (the likes you'll find in any modern game engine). The reason for this is obviously performance. 

Enter the wonderful World of Draw Calls:
Simply put, every time a graphics engine needs to draw an object with a texture-based material (one or multiple textures) on the screen, it sends a message to the CPU to load the object mesh data and the material from the storage medium (conventional hard disk, ssd, DVD,...). 

The CPU then sends the data block to the already waiting GPU, which processes the data block into a visual representation on the screen. 

If you dive further into the matter, you soon find out, that every UV island has seams that produce another draw call per seam, while the GPU figures out how to stitch the islands together. 

Every draw call produces load on CPU, GPU, RAM, VRAM and the storage medium. If you have loads of them, the frame rate will suffer. 

Conclusion 
Again, in highly simplified terms, you get the amount of draw calls per object by adding up the materials and the texture islands on the UV map. 

This is the reason why game asset artists try to use every trick in the book to use the maximal available texture space while maintaining a the lowest amount of UV islands. 

Further Information on Draw Calls
Simon Scheibt's Render Hell delivers the most complete source on the topic that I know of.